1//===-- Implementation of exphk function ----------------------------------===//
2//
3// Part of the LLVM Project, under the Apache License v2.0 with LLVM Exceptions.
4// See https://llvm.org/LICENSE.txt for license information.
5// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception
6//
7//===----------------------------------------------------------------------===//
8
9#include "exphk.h"
10#include "src/__support/CPP/bit.h"
11#include "src/__support/common.h"
12#include "src/__support/fixed_point/fx_bits.h"
13#include "src/__support/macros/config.h"
14
15namespace LIBC_NAMESPACE_DECL {
16
17namespace {
18
19// Look up tables for exp(hi) and exp(mid).
20// Generated with Sollya:
21// > for i from 0 to 89 do {
22// hi = floor(i/8) - 5;
23// m = i/8 - floor(i/8) - 0.5;
24// e_hi = nearestint(exp(hi) * 2^7) * 2^-7;
25// e_mid = nearestint(exp(m) * 2^7) * 2^-7;
26// print(hi, e_hi, m, e_mid);
27// };
28// Notice that when i = 88 and 89, e_hi will overflow short accum range.
29static constexpr short accum EXP_HI[12] = {
30 0x1.0p-7hk, 0x1.0p-6hk, 0x1.8p-5hk, 0x1.1p-3hk, 0x1.78p-2hk, 0x1.0p0hk,
31 0x1.5cp1hk, 0x1.d9p2hk, 0x1.416p4hk, 0x1.b4dp5hk, 0x1.28d4p7hk, SACCUM_MAX,
32};
33
34static constexpr short accum EXP_MID[8] = {
35 0x1.38p-1hk, 0x1.6p-1hk, 0x1.9p-1hk, 0x1.c4p-1hk,
36 0x1.0p0hk, 0x1.22p0hk, 0x1.48p0hk, 0x1.74p0hk,
37};
38
39} // anonymous namespace
40
41LLVM_LIBC_FUNCTION(short accum, exphk, (short accum x)) {
42 using FXRep = fixed_point::FXRep<short accum>;
43 using StorageType = typename FXRep::StorageType;
44 // Output overflow
45 if (LIBC_UNLIKELY(x >= 0x1.64p2hk))
46 return FXRep::MAX();
47 // Lower bound where exp(x) -> 0:
48 // floor(log(2^-8) * 2^7) * 2^-7
49 if (LIBC_UNLIKELY(x <= -0x1.63p2hk))
50 return FXRep::ZERO();
51
52 // Current range of x:
53 // -0x1.628p2 <= x <= 0x1.638p2
54 // Range reduction:
55 // x = hi + mid + lo,
56 // where:
57 // hi is an integer
58 // mid * 2^3 is an integer
59 // |lo| <= 2^-4.
60 // Then exp(x) = exp(hi + mid + lo) = exp(hi) * exp(mid) * exp(lo)
61 // ~ exp(hi) * exp(mid) * (1 + lo)
62 // with relative errors < |lo|^2 <= 2^-8.
63 // exp(hi) and exp(mid) are extracted from small lookup tables.
64
65 // Round-to-nearest 1/8, tie-to-(+Int):
66 constexpr short accum ONE_SIXTEENTH = 0x1.0p-4hk;
67 // x_rounded = floor(x + 1/16).
68 short accum x_rounded = ((x + ONE_SIXTEENTH) >> (FXRep::FRACTION_LEN - 3))
69 << (FXRep::FRACTION_LEN - 3);
70 short accum lo = x - x_rounded;
71
72 // Range of x_rounded:
73 // x_rounded >= floor((-0x1.628p2 + 0x1.0p-4) * 2^3) * 2^-3
74 // = -0x1.6p2 = -5.5
75 // To get the indices, we shift the values so that it start with 0.
76 // Range of indices: 0 <= indices <= 89
77 StorageType indices = cpp::bit_cast<StorageType>((x_rounded + 0x1.6p2hk) >>
78 (FXRep::FRACTION_LEN - 3));
79 // So we have the following relation:
80 // indices = (hi + mid + 44/8) * 8
81 // That implies:
82 // hi + mid = indices/8 - 5.5
83 // So for lookup tables, we can use the upper 4 bits to get:
84 // exp( floor(indices / 8) - 5 )
85 // and lower 3 bits for:
86 // exp( (indices - floor(indices)) - 0.5 )
87 short accum exp_hi = EXP_HI[indices >> 3];
88 short accum exp_mid = EXP_MID[indices & 0x7];
89 // exp(x) ~ exp(hi) * exp(mid) * (1 + lo);
90 return (exp_hi * (exp_mid * (0x1.0p0hk + lo)));
91}
92
93} // namespace LIBC_NAMESPACE_DECL
94

source code of libc/src/stdfix/exphk.cpp